(Un)expected Monetary Policy Shocks and Term Premia

SFB 649 Discussion Paper 2017-015

(Un)expected

Monetary Policy Shocks and Term Premia

Martin Kliem*

Alexander Meyer-Gohde*²

* Deutsche Bundesbank, Germany

*² Universität Hamburg & Humboldt-Universität zu Berlin, Germany

This research was supported by the Deutsche

Forschungsgemeinschaft through the SFB 649 "Economic Risk".

http://sfb649.wiwi.hu-berlin.de ISSN 1860-5664

SFB

6 4 9

E C O N O M I C

R I S K

B E R L I N

(Un)expected Monetary Policy Shocks and Term Premia ^∗

Martin Kliem

Alexander Meyer-Gohde

July 6, 2017

Abstract

We analyze an estimated stochastic general equilibrium model that replicates key macroeconomic and financial stylized facts during the Great Moderation of 1983-2007.

Our model predicts a sizeable and volatile nominal term premium - comparable to recent reduced-form empirical estimates - with real risk two times more important than inflation risk for the average nominal term premia. The model enables us to address salient questions about the effects of monetary policy on the term structure of interest rates. We find that monetary policy shocks can have differing effects on risk premia. Actions by the monetary authority with a persistent effect on households’

expectations have substantial effects on nominal and real risk premia. Our model rationalizes many of the opposing findings on the effects of monetary policy on term premia in the empirical literature.

JEL classification: E13, E31, E43, E44, E52

Keywords: DSGE model, Bayesian estimation, Term structure, Monetary policy

∗We thank Klaus Adam, Martin Andreasen, Michael Bauer, Michael Joyce, Alexander Kriwoluzky, Sydney Ludvigson, Jochen Mankart, Christoph Meinerding, Emanuel M¨onch, St´ephane Moyen, and Harald Uhlig for helpful comments. Moreover, we have benefitted from seminar discussions at the Dynare conference, Shanghai, at Hamburg University, at LMU Munich, at the CEF, New York, and at the Bundesbank, which are gratefully acknowledged. We thank Eric T. Swanson and Michael Bauer for sharing their estimates of the ten year nominal term premium with us. This research was supported by the Deutsche Forschungsgemeinschaft through the CRC 649 Economic Risk. This paper represents the authors’ personal opinions and does not necessarily reflect the views of the Deutsche Bundesbank or its staff. Finally, we thank Andreea Vladu for excellent research assistance.

†Deutsche Bundesbank, Economic Research Center, Wilhelm-Epstein-Str. 14, 60431 Frankfurt am Main, Germany, email: martin.kliem@bundesbank.de

‡Universit¨at Hamburg, Fakult¨at f¨ur Wirtschafts- und Sozialwissenschaften, Von-Melle-Park 5, 20146 Hamburg, email: alexander.mayer-gohde@wiso.uni-hamburg.de

Humboldt-Universit¨at zu Berlin, Institut f¨ur Wirtschaftstheorie II, Spandauer Straße 1, 10178 Berlin, Germany; email: alexander.meyer-gohde@wiwi.hu-berlin.de

1 Introduction

What are the effects of monetary policy on the term structure of interest rates? The empiri- cal literature has yet to reach a definitive conclusion on this question, not only quantitatively but also qualitatively. We contribute to this discussion with an estimated stochastic dynamic general equilibrium model (DSGE) which replicates key macroeconomic and financial stylized facts during the Great Moderation of 1983-2007. In contrast to standard structural mod- elling approaches – like the linear New Keynesian models commonly used in policy analysis – our model captures the impact of monetary policy on interest rates beyond the expecta- tion hypothesis and, therefore, is well positioned to answer our introductory question. We show that different monetary policy actions can have substantially different effects on risk premia. First, unexpected transitory changes of the policy rate have limited effects on nom- inal and real term premia. Second, expected monetary policy shocks, such as unconditional forward guidance, affect households’ future expectation regarding real and nominal variables substantially. This has significant effects on households’ precautionary savings motives and, consequentially, on risk premia in the economy. Similarly, shocks to the inflation target have persistent effects on the systematic behavior of monetary policy, generating strong effects in risk premia. By distinguishing between these different monetary policy actions, our struc- tural model rationalizes many of the opposing findings on the effects of monetary policy on term premia in the empirical literature (see, for example,Hanson and Stein,2015;Nakamura and Steinsson, 2017).

A comprehensive analysis of monetary policy needs a quantitative structural model that captures the nonlinearity behind the risk of underlying financial variables and simultaneously replicates key stylized macroeconomic facts. However, as poignantly phrased by G¨urkaynak and Wright (2012, p. 354): “A general problem with a structural model [. . .] is that it is challenging to maintain computational tractability and yet obtain time-variation in term premia.” We address this problem and estimate a New Keynesian macro-finance model with U.S. data from 1983:Q1 to 2007:Q4 using a new and computationally efficient procedure that captures both constant and time varying risk premia by maintaining linearity in states and shocks (Meyer-Gohde, 2016). This approach allows us to investigate a structural model in the spirit of Smets and Wouters (2003, 2007) and Christiano, Eichenbaum, and Evans (2005), and is able to provide not only an in-depth analysis of the macroeconomy but also of the term structure of interest rates and their interactions. Figure1shows that our structural model predicts a historical 10-year term premium comparable in level, pattern, and volatility with recent reduced-form empirical estimates.¹ Our model predicts both an upward sloping

1The gray area in Figure1 presents the range (maximum and minimum) of the estimates for the 10-year term premium from models developed by Kim and Wright(2005), Rudebusch and Wu (2008), Bernanke,

1985 1990 1995 2000 2005 Year

0 1 2 3 4 5 6 7

Nominal 10-year term premium in %

Model

Corresponding estimates in the literature

Figure 1: Model implied 10-year nominal term premium (black line) and range of corre- sponding estimates in the literature (gray area).

nominal yield curve in line with the data and an upward sloping real yield curve in line with empirical estimates (see, for example, G¨urkaynak, Sack, and Wright, 2010; Chernov and Mueller,2012). Our real yield curve is in contrast to many DSGE models (see, for example, van Binsbergen, Fern´andez-Villaverde, Koijen, and Rubio-Ram´ırez, 2012; Swanson, 2016) that generally attribute a stronger insurance-like character to real bonds leading to flat or downward sloping real yield curves. Additionally, our results suggest that 2/3 of the average slope of the nominal term structure is related to real rather than to inflation risk. In this regard, the model implied upward sloping inflation risk premium is consistent with recent estimates in the literature (see, for example, Abrahams, Adrian, Crump, Moench, and Yu, 2016), with our average term structure of inflation risk comfortably between the estimates of Buraschi and Jiltsov (2005) and Chen, Liu, and Cheng (2010). In summary, our model- implied estimates demonstrate a considerable alignment with various empirical estimates in the literature. This alignment is all the more remarkable as these measures, with the exception of nominal yields, were not used in our estimation. This provides us with a high degree of confidence in our model as we proceed to the structural analysis of the effects of monetary policy on the term structure of interest rates and its components.

Reinhart, and Sack (2004),Adrian, Crump, and Moench(2013), andBauer(2016). The first three measures were calculated by Rudebusch, Sack, and Swanson (2007) and Rudebusch, Swanson, and Wu (2006). A description of the estimates can be found there. We are very thankful to Eric T. Swanson and Michael Bauer for sharing their estimates with us.

Our structural analysis contributes to the growing body of empirical investigations into the effects of conventional and, more recently unconventional, monetary policy on the term structure of interest rates.² So far the empirical literature disagrees not only on the quantita- tive effects of monetary policy shocks on term premia (seeHanson and Stein,2015;Nakamura and Steinsson, 2017), but also on their qualitative effects - i.e., whether interest rates and term premia comove (see Abrahams et al., 2016; Crump et al., 2016). There are many po- tential reasons for this lack of robustness and an analysis of them is beyond the scope of this paper.³ Instead, we take our cue from Ramey(2016) who notes that the “shocks” identified in the empirical literature are not always the empirical counterparts of shocks from theo- retical models. For example, with monetary policy following a Taylor-type rule, we want to disentangle changes in the systematic behavior of monetary policy - due, for example, to changes in the inflation target - from innovations to the Taylor rule and from preannounced monetary actions like forward guidance (see, for example, Woodford,2012).

We find that an unexpected monetary policy shock via a simple innovation to the Taylor rule affects risk premia at shorter more strongly than longer maturities (see Nakamura and Steinsson, 2017, for a comparable emiprical finding), but overall has limited effects on the term premia at all maturities. This finding is in line with those of other structural models (see, for example,Rudebusch and Swanson,2012) and confirms some of the empirical findings of Nakamura and Steinsson (2017). Simply put, an uncorrelated innovation to the Taylor rule dies out too quickly to have substantial effects at business cycle frequencies. Therefore, the effects on risk premia, which vary primarily at lower frequencies (see, for example, Piazzesi and Swanson, 2008), are limited. In contrast, a shock to the inflation target has much stronger effects on the term structure of interest rates across all maturities. The reason behind the strong effect on the risk premia, as laid out byRudebusch and Swanson(2012), is that a change to the inflation target introduces long-run (nominal) risk which is per se longer lasting and so has stronger effects on households’ expectation formation, their precautionary savings motives and, thus, on risk premia. The strong quantitative effects of such a monetary action are comparable to the findings of Hanson and Stein (2015). Additionally for longer maturities, the policy rate and risk premia comove on impact (see, for example, Hanson and Stein, 2015; Abrahams et al., 2016) after such a more systematic change of monetary policy. Contrarily, we find for a simple innovation to the Taylor rule that risk premia for long

2See for example the pioneering work byKuttner(2001),Cochrane and Piazzesi(2002), andG¨urkaynak, Sack, and Swanson(2005a,b). More recent papers that also place a focus on unconventional monetary policy are, for example,Nakamura and Steinsson (2017),Gertler and Karadi (2015),Gilchrist, L´opez-Salido, and Zakrajˇsek (2015),Abrahams et al.(2016), andCrump, Eusepi, and Moench(2016).

3For example, different underlying samples or identification approaches could be to blame. See, for example, Campbell, Fisher, Justiniano, and Melosi (2016) for a discussion of potential shortcomings in isolating the effects of monetary policy in the recent literature.

maturities tend to move opposite the policy rate on impact, i.e., a looser monetary policy increases risk premia. In particular in our model, such a looser monetary policy increases the precautionary savings motive of agents as they expect more volatile inflation and output and, therefore, demand higher risk premia. This finding is comparable to the empirical results of Crump et al. (2016) and Nakamura and Steinsson (2017).

Following the approach of Woodford (2012), we analyze the effects of unconditional for- ward guidance.⁴ This is accomplished by adding a sequence of anticipated shocks to the Taylor rule to keep the policy rate upon announcement constant until the announced inter- est rate change (here a cut) is implemented. We find that this kind of forward guidance affects risk premia substantially, prying bond yields from the expectations hypothesis. In particular, we find that a commitment to a future reduction in the policy rate and constant policy rates until then causes real term premia and inflation risk premia to rise as agents expect more volatile inflation and output in the future. This finding is in line with the empirical finding of Akkaya et al. (2015). Turning to the inflation risk premia, its increase follows what theory would predict: While forward guidance does communicate the expected path of future short rate, it is just as informative about the central bank’s commitment to allow higher inflation in the future. This mechanism increases households’ precautionary savings motives and their demand for higher inflation risk premia.

The reminder of the paper reads as follows: Section 2 presents the model. Following, section 3 describes the solution method, the data, and the Bayesian estimation approach in greater detail. Section 4 presents the estimation results and discusses the model fit. Section 5 presents the effects of unexpected and expected monetary policy on the term structure.

Section 6 concludes the paper.

2 Model

In the following section, we present our dynamic stochastic general equilibrium (DSGE) model. We study a New Keynesian model, in which households have recursive preferences following Epstein and Zin (1989, 1991) and Weil (1989), maximize their utility from con- sumption relative to a habit and labor, and accumulate capital. The nominal yield curve is derived from the households’ stochastic discount factor and no-arbitrage restrictions. Firms are monopolistic competitors selling differentiated products at prices that are allowed to adjust in a stochastic fashion as in Calvo (1983). The central bank follows a Taylor rule

4For a discussion of different forms of forward guidance seeCampbell, Evans, Fisher, and Justiniano(2012) and Akkaya, G¨urkaynak, Kısacıko˘glu, and Wright (2015). Particularly such a distinction is a significant challenge in many empirical approaches (see, for example, the discussion inNakamura and Steinsson,2017;

Campbell et al.,2016).

which sets the short-term nominal interest rate as a function of the inflation rate and output.

The model has a similar structure to Smets and Wouters (2003,2007) and Christiano et al.

(2005) by including nominal and real rigidities which have demonstrated success in repli- cating stylized facts of the macroeconomy. Additionally, the model incorporates real and nominal long-run risk (Bansal and Yaron, 2004; G¨urkaynak et al., 2005b) which, together with recursive preferences, have been highlighted in the literature as important in order to explain many financial moments in consumption-based asset pricing models.

2.1 Firms

A perfect competitive representative firm produces the final good yt. This final good is an aggregate of a continuum of intermediate goods yj,t and given by the function

y_t= Z 1

0

y

θp−1 θp

j,t dj 1−^θpθp

(1) with θp >1 the intratemporal elasticity of substitution across the intermediate goods. The competitive, representative firm takes the price of output,P_t, and the price of inputs, P_t(j), as given. The resulting demand function for the intermediate good is

yj,t = Pj,t

P_t −θp

yt (2)

and the aggregate price level is defined as

Pt= Z 1

0

P_j,t^1−θpdj _1−θp¹

(3) and gross inflation is πt=Pt/Pt−1.

The intermediate goodj is produced by a monopolistic competitive firm with the follow- ing Cobb-Douglas production function

yj,t = exp{at}k_j,t−1^α (ztlj,t)^1−α−z_t⁺Ωt (4) wherekj,t andlj,t denote capital services and the amount of labor used for production by the jth intermediate good producer, respectively. The parameterαdenotes the output elasticity with respect to capital and Ωt the fixed costs of production. The variable exp{at} refers to

a stationary technology shock, where a_t is described by the following AR(1) process at =ρaat−1 +σaa,t, with a,t

iid∼ N(0,1) (5)

The variable zt depicts a stochastic aggregate productivity trend. We include this non- stationary productivity shock to allow for a source of real long-run risk. As put forward by Bansal and Yaron (2004), the presence of real long-run risk is important in order to explain many financial moments in a consumption-based asset pricing model. We assume that exp{µ_z,t}=z_t/z_t−1 and let

µ_z,t= (1−ρ_z) ¯µ_z+ρ_zµ_z,t−1+σ_zz,t, with _z,t^iid∼ N(0,1) (6) The economy has two sources of growth. Alongside the stochastic trend in productivity zt, the economy also faces a deterministic trend in the relative price of investment Υt with exp{µ¯_Υ}= Υ_t/Υ_t−1. We followAltig, Christiano, Eichenbaum, and Linde(2011) and define z_t⁺ = Υ

α 1−α

t zt, which can be interpreted as an overall measure of technological progress in the economy. The overall trend in the economy is characterized by

µ_z⁺_,t = α

1−αµ¯_Υ+µ_z,t (7)

Finally, we scale Ωt byz⁺_t to ensure the existence of a balanced growth path and let produc- tion costs be time-varying as proposed by Andreasen(2011). In our model, such variations in firms’ fixed production costs represent real supply shocks by assuming that

log Ω_t

Ω¯

=ρΩlog

Ω_t−1 Ω¯

+σΩΩ,t, with Ω,t

iid∼ N(0,1) (8)

Following Calvo (1983), intermediate good firms are subject to staggered price setting, i.e., they are allowed to adjust their prices only with probability (1−γp) each period. If a firm cannot re-optimize, its price evolves according to the indexation rule: Pj,t =Pj,t−1π^ξ_t−1^p . When the firm is able to optimally adjust its price, the firm sets the price ˜p_t = P_j,t to maximize the value of its expected future dividend stream subject to the demand it faces and taking into account the indexation rule and the probability of not being able to readjust.

The first order conditions of this maximization problem are

K^t =ytp˜^−θ_t ^p+γpEt



M_t+1 π^ξ_t^p π_t+1

!1−θp

˜ pt

˜ p_t−1

−θp

K^t+1



 (9)

and

θp−1

θp K^t=ytmctp˜^−θ_t ^p−1+γpEt



M_t+1 π_t^ξp πt+1

!−θp

˜ pt

˜ pt−1

−θp−1

θp−1 θp K^t+1



 (10)

which is the same for all firms that can adjust their price in periodt. Moreover, the variable Mt+1 represents the real stochastic discount factor of the representative household from periodt to t+ 1 and mct the real marginal costs of the intermediate good firm. In sum, the aggregate price index evolves according to

1 =γp

π_t−1^ξp πt

!1−θp

+ (1−γp) (˜pt)^1−θp (11)

2.2 Households

We assume that the representative household has recursive preferences as postulated by Epstein and Zin (1989, 1991) and Weil (1989). Following Rudebusch and Swanson (2012), the value function of the household can be written as

Vt=





ut+β Et

V_t+1^1−σEZ_1−σEZ¹

if ut >0 for all t ut−β Et

(−Vt+1)^1−σEZ_1−σEZ¹

if ut <0 for all t (12) whereutis the household’s period utility kernel andβ ∈(0,1) the subjective discount factor.

Forσ_EZ >0, these preferences allow us to disentangle the household’s risk aversion from its intertemporal elasticity of the substitution (IES). For σ_EZ = 0, eq. (12) reduces to standard expected utility.

Similarly to Andreasen, Fern´andez-Villaverde, and Rubio-Ram´ırez (2017), the utility kernel takes the following functional form

ut= exp{εb,t}

"

1 1−γ

ct−bht

z_t⁺

1−γ

−1

!

+ ψL

1−χ(1−lt)^1−χ

#

(13) with consumption ct, the predetermined stock of consumption habits ht, hours worked lt, and preference parameters γ, χ, and ψL. The habit stock is external to the household, thus we set ht = Ct−1, the level of aggregate consumption in the previous period. The parameter b ∈ (0,1) controls the degree of external habit formation. The presence of habit formation enables the model to match macroeconomic as well as asset pricing moments jointly as discussed in the literature (see, for example, H¨ordahl, Tristani, and Vestin, 2008;

van Binsbergen et al.,2012). The variable exp{ε_b,t}represents a preference shock, whereε_b,t evolves according to the process

εb,t=ρbεb,t−1+σbb,t, with b,t

iid∼N(0,1) (14)

As described above, the variable z_t⁺ represents the overall level of technology in the econ- omy and, by expressing habit-adjusted consumption relative to this trend, the utility kernel ensures a balanced growth path (see, for example, An and Schorfheide, 2007).

The household’s real period-by-period budget constraint reads

ct+ It

Υt

+bt+Tt=wtlt+r_t^kkt−1+

bt−1expn R^f_t−1o πt

+ Z 1

0

Πt(j)dj (15) where the left-hand side represents the household’s resources spent on consumption, invest- ment It, a lump-sum tax Tt, and a one-period bond bt that accrues the risk-free nominal interest R^f_t in the following period. The right-hand side of eq. (15) describes the income of the household in periodt. It consists of labor incomewtltwithwtthe real wage, income from capital services sold to firms last period r^k_tkt−1, the pay-off from bonds issued one period beforebt−1. Finally, the term Π (j) represents the income from dividends of monopolistically competitive intermediate firms – indexed by j – owned by households.

The households own the economy-wide physical capital stock, which accumulates accord- ing to the following law of motion

kt = (1−δ)kt−1+ exp{εi,t} 1− ν 2

It

It−1 −exp{µ¯z⁺+ ¯µΥ} 2!

It (16) where δ is the depreciation rate and ν ≥ 0 introduces investment adjustment costs as in Christiano et al. (2005). The term exp{µ¯_z⁺ + ¯µ_Υ} ensures that the investment adjustment costs are zero along the balanced growth path. Following Justiniano, Primiceri, and Tam- balotti (2010), the variable exp{εi,t} represents an investment shock which measures the exogenous variation in the efficiency with which the final good can be transformed into physical capital and thus into tomorrow’s capital input, where εi,t evolves according to the process:

εi,t =ρiεi,t−1+σii,t, with i,t

iid∼ N(0,1) (17)

2.3 Monetary Policy

We follow Rudebusch and Swanson (2008, 2012) and assume that monetary policy sets the one-period nominal interest rateR^f_t by following a Taylor-type policy rule expressed annually

4R^f_t = 4·ρRR^f_t−1+(1−ρR)

4¯r^real+ 4 logπt+ηylog yt

z_t⁺y¯

+ηπlog πt

π_t^∗

+σmm,t (18) where ¯r^real is the real interest rate at the deterministic steady state and ρR, ηy, and ηπ are policy parameters that characterize the systematic response of the central bank. The term m,t represents a shock to the nominal interest rate which is assumed to be iid normally distributed with mean 0 and variance 1. Monetary policy aims to stabilize the inflation gap, log (πt/π_t^∗), and the output gap, log yt/z_t⁺y¯

. The output gap is characterized by the deviation of actual output from its balanced growth path. The inflation gap is characterized by the deviation of inflation from the central bank’s inflation target π_t^∗. Rudebusch and Swanson(2012) interpret changes in the inflation target as long-run nominal (inflation) risk and show that the existence of such long-run risk is helpful in explaining the historical U.S.

term premium. We follow G¨urkaynak et al. (2005b) and Rudebusch and Swanson (2012) and assume that the inflation target is time-varying and is described by the following law of motion

logπ_t^∗−4 log ¯π =ρπ logπ^∗_t−1−4 log ¯π

+ 4ζπ(logπt−1−log ¯π,) +σππ,t (19) with _π,t representing a shock to the inflation target, assumed iid normal with mean 0 and variance 1.

2.4 Aggregation and Market Clearing

The aggregate resource constraint in the goods market is given by

p⁺_tyt= exp{at}k_t−1^α (ztlt)^1−α−z_t⁺Ωt (20) where l_t = R1

0 l(j, t) dj and k_t = R1

0 k(j, t) dj are the aggregate labor and capital inputs, respectively. The term p⁺_t = R1

0

Pj,t

Pt

−θp

dj measures the price dispersion arising from staggered price setting. Price distortion follows the law of motion

p⁺_t = (1−γ_p) (˜p_t)^−θp+γ_p π_t−1^ξp πt

!−θp

p⁺_t−1 (21)

Finally, the economy’s aggregate resource constraint implies that yt=ct+ It

Υ_t +gt (22)

whereg_t= ¯gz_t⁺exp{ε_g,t}represents government consumption expenditures, which are grow- ing with the economy and are financed by lump-sum taxes g_t = T_t. The variable exp{ε_g,t} represents an exogenous shock to government consumption with εg,t evolving according to the following AR(1) process

εg,t=ρgεg,t−1+σgg,t, with g,tiid

∼N(0,1) (23)

2.5 The Nominal and Real Term Structures

The derivation of the nominal and real term structure in our model is identical to the procedure described, for example, by Rudebusch and Swanson (2008, 2012) and Andreasen (2012a). Specifically, the price of any financial asset equals the sum of the stochastically discounted state-contingent payoffs of the asset in periodt+1 following standard no-arbitrage arguments. For example, the price of a default free n-period zero-coupon bond that pays one unit of cash at maturity satisfies

Pn,t=Et

M_t,t+n^$ 1

(24)

=Et

M_t,t+1^$ Pn−1,t+1

where M_t,t+1^$ is the household’s nominal stochastic discount factor, which has the following functional form

M_t,t+1^$ =β λt+1

λ_tπ_t+1 (Vt+1)^−σEZEt

V

σEZ 1−σEZ

t+1

(25) with λt the marginal utility of consumption. Additionally, the continuously compounded yield to maturity on the n-period zero-coupon nominal bond is defined as

exp

−nR^$_n,t =P_n,t^$ (26) Following the literature (e.g. Rudebusch and Swanson, 2012), we define the term premium on a long-term bond as the difference between the yield on the bond and the unobserved risk-neutral yield for that same bond. Similarly to eq. (24), this risk-neutral bond price, ˆPn,t, which pays also one unit of cash at maturity, is defined as

Pˆn,t = expn

−R_t^fo Et

hPˆn−1,t+1

i (27)

In contrast to eq. (24), discounting is performed using the risk-free rate (with R^f_t equal to the expression R_1,t) rather than the stochastic discount factor. Accordingly, the nominal term premium on a bond with maturity n is given by

T P^$_n,t = 1 n

log ˆPn,t−logP_n,t^$

(28) Similarly, we can derive the yield to maturity of a real bond R_n,t as well a the price of risk-neutral real bond. Hence, it is straightforward to solve also for the real term premium T Pn,t of a bond with maturity n. Finally, we follow the literature and define inflation risk premia T P_n,t^π in our model as

T P_n,t^π =T P_n,t^$ −T Pn,t (29)

3 Model Solution and Estimation

3.1 Solution Method

We adopt the method of Meyer-Gohde (2016) to solve the model. This approximation ad- justs a linear in states approximation for risk and provides derivations for the approximation around the means of the endogenous variables approximated out to a finite moment of the underlying stochastic driving forces.⁵ This allows us to use the standard set of macroecono- metric tools for estimation and analysis of linear models, without limiting the approximation to the certainty-equivalent approximation around the deterministic steady state. We adjust the points and slopes of the decision rules for risk out to the second moments of the underly- ing stochastics to capture both constant and time-varying risk premium, as well as the effects of conditional heteroskedasticity (e.g. van Binsbergen et al., 2012). Unlike standard higher order polynomial perturbations⁶ or affine approximation methods,⁷ this linear in states ap- proximation gives us significant computational advantages for iterative calculations such as the Metropolis-Hastings algorithm we will use to sample from the posterior distribution of

5Meyer-Gohde(2016) provides derivations for adjustments around the deterministic and stochastic steady states, along with those around the mean that we derive and apply here, accuracy checks and formal justi- fications for the method.

6Among others, recent third order perturbation approximations for DSGE models of the term struc- ture include Rudebusch and Swanson (2008, 2012), van Binsbergen et al. (2012) Andreasen (2012a), and Andreasen et al.(2017).

7These approaches separate the macro and financial variables, generally using a (log) linear approximation of the former and an affine approximation for the yield curve following the empirical finance literature. Bonds are priced in an arbitrage free setup using either the endogenous pricing kernel implied by households’

stochastic discount factors, asDew-Becker(2014),Bekaert, Cho, and Moreno(2010), andPalomino(2012), or an estimated exogenously specified kernel, asH¨ordahl, Tristani, and Vestin(2006) ,H¨ordahl and Tristani (2012),Ireland(2015),Rudebusch and Wu(2007),Rudebusch and Wu(2008).

the parameters while maintaining the endogenous pricing of risk implied by agents’ optimiz- ing behavior. Appendix B provides a self-contained overview of the derivations involved in this approximation.

The tension between the nonlinearity need to capture the time varying effects of risk underlying asset prices on the one hand and the difficulties bringing nonlinear estimation routines such as the particle filter to bear on such models on the other is highlighted by van Binsbergen et al. (2012), who model inflation as exogenous in a New Keynesian model to make their Bayesian likelihood estimation tractable. The advantage of a linear in state approximation for estimation has also been noted by, e.g., Ang and Piazzesi(2003),Hamil- ton and Wu (2012), Dew-Becker (2014). Our approach compromises between the goals of nonlinearity in risk to capture financial variables and the endogenous stochastic discount factor to price financial variables consistent with the macroeconomy on the one hand, and the need for linearity in states to make the estimation of medium scale policy relevant mod- els feasible on the other. To further reduce the computational burden, we apply the PoP method ofAndreasen and Zabczyk(2015) that solves the model in a two-step fashion. First, the policy rules for the macro side, including the pricing kernel and the nominal short rate, are approximated and then the financial variables are solved for using this policy function.

It is important to note that this is not a further approximation, but rather the recognition that the equations that price different maturities such as eq. (24) are forward recursions that do not enlarge the state space.

3.2 Data

We estimate the model with quarterly U.S. data from 1983:q1 to 2007:q4. Thus, our sample covers the Great Moderation, stopping right before the onset of the Great Recession. This period is chosen specifically for two reasons. First, it is widely accepted in the literature that the U.S. faced a systematic change in monetary policy after Paul Volcker became chairman of the Federal Reserve (e.g. Clarida, Gal´ı, and Gertler, 2000). Second, the start of the Great Recession, the financial crisis of 2008, along with the zero interest policy rates that prevailed from December 2008 onward marks another structural change in U.S. monetary policy. While the systematic behavior of monetary policy is an important driver of the yield curve, as pointed out, for example, by Rudebusch and Swanson (2012), we chose a time episode which is characterized by a relatively stable monetary policy regime.⁸

Our estimation is based on four macroeconomic time series complemented by six time series from the nominal yield curve and two time series of survey data on interest rate

8See, for example,Bikbov and Chernov(2013) andBianchi, Kung, and Morales(2016) for an investigation of policy regime changes and the term structure of interest rates.

forecasts.⁹ The macroeconomic dynamics are characterized by real GDP growth, real private investment growth, real private consumption growth, and annualized GDP deflator inflation rates. While the last is measured in levels, the remaining variables are expressed in per capita log-differences using the civilian noninstitutional population over 16 years (CNP16OV) series from the U.S. Department of Labor, Bureau of Labor Statistics.

The nominal yield curve is measured by the 1-quarter, 1-year, 3-year, 5-year, and 10- year annualized interest rates of U.S. Treasury bonds. With the exception of the 1-quarter interest rate, the data are from Adrian et al. (2013) which are identical to the otherwise often used time series by G¨urkaynak, Sack, and Wright (2007). For the 1-quarter maturity, we use the 3-month Treasury Bill rate from the Board of Governors of the Federal Reserve System. To have a consistent description of the yield curve, we use this interest rate as the policy rate (R^f_t =R^$_1,t ) in our model instead of the effective Fed funds rate.

Survey data on interest rate forecasts have shown to be helpful to improve the identifi- cation of term structure models (see, for example, Kim and Orphanides, 2012; Andreasen, 2011). For this reason, we incorporate 1 and 4-quarter ahead expectations of the 3-month Treasury Bill into the estimation. The data are taken from the Survey of Professional Fore- casters.

3.3 Bayesian Estimation

In this subsection, we present the prior choices for the estimated parameters as well as the calibration of the parameters we choose not to estimate.

Given the choice of our observable variables and the characteristics of our model, for example, the highly stylized labor market, some of the model parameters can hardly be expected to be identified. These parameters are calibrated either following the literature or related to our observables. In particular, we calibrate the steady state growth rates, ¯z⁺ and Ψ to 0.54/100 and 0.08/100 which implies growth rates of 0.54 and 0.62 percent for GDP and¯ investment as in our sample. Moreover, we calibrate the capital depreciation rate, δ, to 10%

per year and the share of capital, α, in the production function to 1/3. We also assume that in the deterministic steady state, the labor supply ¯l and government consumption to GDP ratio ¯g/¯y are 1/3 and 0.19, respectively. The discount rate β is set equal to 0.99 and the steady state of the elasticity of substitution between the intermediate goods θ_p is equal to 6, implying a markup of 20%. Following Andreasen et al. (2017), we set the price indexation ξp = 0 and calibrate the Frisch elasticity of labor supply F E to 0.5. Hence, we can solve recursively for χ= 1/F E· 1/¯l−1

. Table1 summarizes the parameter calibration.

9See AppendixCfor details on the source and a description of all data used in this paper.

Description Symbol Value

Technology trend in percent z¯⁺ 0.54/100

Investment trend in percent Ψ¯ 0.08/100

Capital share α 1/3

Depreciation rate δ 0.025

Price markup θ_p/(θ_p−1) 1.2

Price indexation ξ_p 0

Discount factor β 0.99

Frisch elasticity of labor supply F E 0.5

Labor supply ¯l 1/3

Ratio of government consumption to output ¯g/¯y 0.19 Table 1: Parameter calibration.

The remaining parameters of the model are estimated. Since the focus of the paper is to jointly explain macroeconomic and asset pricing facts, we pay special attention to selected first and second moments when estimating the DSGE model. As described in Kliem and Uhlig (2016), the practical problem boils down to having just one observation on the means, e.g., of the slope, curvature, and level of the yield curve, while there are many observations to identify parameters crucial for the macroeconomic dynamics of the model. To mitigate this imbalance, we apply an endogenous prior approach similar toDel Negro and Schorfheide (2008) and Christiano, Trabandt, and Walentin(2011). In particular, we use a set of initial priors, p(θ), where the priors are independent across parameters. Then, we use two sets of first and second moments from a pre-sample.¹⁰ We treat the first and second moments of interest separably in two blocks to capture potentially different precisions of beliefs regarding first and second moments. Finally, the product of the initial priors, the likelihood of selected first moments, and the likelihood of selected second moments forms the endogenous prior distribution which we use for the estimation of the model. In the subsequent paragraphs, we describe the method of endogenously formed priors regarding first and second moments as well as its practical application in the paper.

Following Del Negro and Schorfheide (2008), we assume ˆF to be a vector that collects the first moments of interest from our pre-sample and FM (θ) be a vector-valued function which relates model parameters and ergodic means

Fˆ =FM(θ) +η (30)

10In practice, we followChristiano et al.(2011) and use the actual sample as our pre-sample as no other suitable data is available because of the monetary regime changes immediately before and after our sample.

where η is a vector of measurement errors. In our application, we assume that the error terms η are independently and normally distributed. Hence, we express eq. (30) as a quasi- likelihood function which can be interpreted as the conditional density

L

FM(θ)|F , Tˆ ^∗

= exp

−T^∗ 2

Fˆ−FM (θ)0

Σ⁻¹_η

Fˆ−FM(θ)

(31)

=p

Fˆ|FM(θ), T^∗

This quasi-likelihood is small for values of θ that lead the DSGE model to predict first mo- ments that strongly differ from the measures of the pre-sample. The parameter T^∗ captures, along with the standard deviation ofη, the precision of our beliefs about the first moments.

In practice we set T^∗ to the length of the pre-sample.

For the application in this paper, we assume that the vector ˆF contains the mean of inflation and the means of proxies for the level, slope, and curvature factors of the yield curve. We include the mean of inflation because the non-linearities in our model impose strong precautionary motives that push the predicted ergodic mean of inflation away from its deterministic steady state, ¯π, as is also discussed by Tallarini (2000) and Andreasen (2011). Regarding L

FM(θ)|Fˆ

, we assume that Et[400π|θ] is normally distributed with mean 2.5 and variance 0.1.

We follow, e.g., Diebold, Rudebusch, and Aruoba (2006) and specify common proxies for the level, slope, and curvature factors of the yield curve. Specifically, the proxy for the level factor is R^$_1,t+R^$_8,t+R^$_40,t

/3, with all yields expressed in annualized terms and the nominal yield of the 1-quarter Treasury Bond equal to the policy rate in the model.

Additionally, the proxies for the slope and curvature factors are defined as R^$_1,t−R^$_40,t and 2R^$_8,t−R^$_1,t−R^$_40,t, respectively. RegardingL

F_M(θ)|Fˆ

, we assume that the ergodic mean of each factor is normally distributed, with the mean equal to its empirical counterpart of the pre-sample. Moreover, we assume that the means of level, slope, and curvature have a variance of 22, 12, and 9 basis points respectively. Thus, the means and variances can be interpreted as ˆF value and the variance of the measurement error η in eq. (30).

Additionally, we use the second moments of macroeconomic variables, about which we have a priori knowledge, to inform our prior distribution and apply the approach of Chris- tiano et al. (2011). This approach uses classical large sample theory to form a large sample approximation to the likelihood of the pre-sample statistics. The approach is conceptually similar to the one proposed by Del Negro and Schorfheide (2008), but differs in some im- portant respects. Specifically,Del Negro and Schorfheide(2008) focus on the model-implied p-th order vector autoregression, which implies that the likelihood of the second moments

is known exactly conditional on the DSGE model parameters and requires no large-sample approximation in contrast to the approach by Christiano et al. (2011). Yet, the latter ap- proach is more flexible insofar as the statistics to target are concerned. Accordingly, let S be a column vector containing the second moments of interest, then, as shown by Christiano et al. (2011) under the assumption of large sample, the estimator of S is

Sˆ∼N S⁰,ΣˆS

T

!

(32)

with S⁰ the true value of S,T the sample length, and ˆΣS the estimate of the zero-frequency spectral density. Now, let SM(θ) be a function which maps our DSGE model parameters θ into S. Then, for n targeted second moments and sufficiently large T, the density of ˆS is given by

p Sˆ|θ

= T

2π

ⁿ₂ ΣˆS

−¹

2 exp

−T 2

Sˆ−SM(θ)0

Σˆ⁻¹_S

Sˆ−SM(θ)

(33) In our application, S is a set of variances of macroeconomic variables (GDP growth, con- sumption growth, investment growth, inflation, and the policy rate). In sum, the overall endogenous prior distribution takes the following form

p

θ|F ,ˆ S, Tˆ ^∗

=C⁻¹p(θ)p

Fˆ|FM(θ), T^∗ p

Sˆ|θ

(34) where p(θ) is the initial prior distribution and C a normalization constant. Two points are noteworthy. First, while the initial priors are independent across parameters, as is typical in Bayesian analysis, the endogenous prior is not independent across parameters. Second, the normalization constant C is necessary for, e.g., posterior odds calculation but not for estimating the model. Accordingly, we do not calculate this constant, which has otherwise to be approximated (see, for example, Del Negro and Schorfheide, 2008; Kliem and Uhlig, 2016). So, the posterior distribution is given by

p

θ|X,F ,ˆ S, Tˆ ^∗

∝p

θ|F ,ˆ S, Tˆ ^∗

p(X|θ) (35)

with p(X|θ) the likelihood of the data conditional on DSGE model parameters θ.

Table 2 summarizes the initial prior distributions of the remaining parameters. While the prior distributions for most of the parameters are chosen following the literature, it is noteworthy to highlight some deviations. First, we do not use a prior for the preference parameters, γ and αEZ, directly, but rather impose priors for the intertemporal elasticity

Name Symbol Domain Density Para(1) Para(2)

Relative risk aversion RRA/100 R⁺ Uniform 0 20

Calvo parameter γ_p [0,1) Beta 0.5 0.1

Investment adjustment ν R⁺ Gamma 4.0 0.75

Habit formation b [0,1) Beta 0.5 0.1

Intertemporal elas. substitution IES [0,1) Beta 0.25 0.1

Steady state inflation 100 (¯π−1) R⁺ Uniform 0 6

Interest rate AR coefficient ρR [0,1) Beta 0.8 0.1

Interest rate inflation coefficient ηπ R⁺ Gamma 1 0.15

Interest rate output coefficient ηy R⁺ Gamma 0.5 0.1

Inflation target coefficient 100ζπ [0,1) Beta 0.3 0.1

AR coefficient technology ρ_a [0,1) Beta 0.75 0.1

AR coefficient preference ρ_b [0,1) Beta 0.75 0.1

AR coefficient investment ρ_i [0,1) Beta 0.75 0.1

AR coefficient gov. spending ρ_g [0,1) Beta 0.75 0.1

AR coefficient inflation target ρ_π [0,1) Beta 0.95 0.025 AR coefficient long-run growth ρz [0,1) Beta 0.75 0.1

AR coefficient fixed costs ρΩ [0,1) Beta 0.75 0.1

S.d. technology 100σa R⁺ InvGam 0.5 2

S.d. preference 100σb R⁺ InvGam 0.5 2

S.d. investment 100σi R⁺ InvGam 0.5 2

S.d. monetary policy shock 100σm R⁺ InvGam 0.5 2

S.d. government spending 100σ_g R⁺ InvGam 0.5 2

S.d. inflation target 100σ_π R⁺ InvGam 0.06 0.03

S.d. long-run growth 100σ_z R⁺ InvGam 0.5 2

S.d. fixed costs 100σ_Ω R⁺ InvGam 0.5 2

ME 1-year T-Bill 4R^$_4,t R⁺ InvGam 0.005 ∞

ME 2-year T-Bill 4R^$_8,t R⁺ InvGam 0.005 ∞

ME 3-year T-Bill 4R^$_12,t R⁺ InvGam 0.005 ∞

ME 5-year T-Bill 4R^$_20,t R⁺ InvGam 0.005 ∞

ME 10-year T-Bill 4R^$_40,t R⁺ InvGam 0.005 ∞

ME 1Q-expected policy rate 4Et

h R^f_t,t+1i

R⁺ InvGam 0.005 ∞

ME 4Q-expected policy rate 4E_th R^f_t,t+4i

R⁺ InvGam 0.005 ∞

Table 2: Initial prior distribution. Para(1) and Para(2) correspond to means and standard deviations for the Beta, Gamma, Inverted Gamma, and Normal distributions and to the lower and upper bounds for the Uniform distribution.

of substitution, IES, and the coefficient relative risk aversion, RRA, and solve for the underlying parameters. The intertemporal elasticity of substitution, IES, in our model with external habit formation is

IES = 1 γ

1− b

exp (¯z⁺)

(36) We followSwanson(2012) by using his closed-form expressions for risk aversion,RRA, which takes into account that households can vary their labor supply. Hence, our model implies

RRA= γ

1−_exp(¯^b_z⁺₎ +_χ^γ 1−¯l_w¯

¯ c

+α_EZ 1−γ

1− _exp(¯^b_z⁺₎−

1− _exp(¯^b_z⁺₎γ

¯

c^γ−1+^w¯(^1−¯l)

¯ c

1−γ 1−χ

(37)

where ¯l is the steady state labor supply, while ¯c and ¯w are consumption and the real wage in the deterministic steady state, respectively. Given the wide range of different estimates for relative risk aversion in the macro- and finance literatures, we initially assume a uniform prior with support over the interval 0 to 2000; our endogenous prior approach, however, does impose an informative prior. We proceed analogously for the deterministic steady state of inflation and choose an uninformative initial prior distribution. Finally, we add measurement errors to the 1-year, 2-year, 3-year, 5-year, and 10-year Treasury bond yields as well as to the expected policy rate expected 1 and 4-quarters ahead. By adding measurement errors along the yield curve, we are following the empirical term structure literature (see, for example, Diebold et al.,2006) and the measurement errors on the expectations of the short rate align the imperfect fit of the data with the model’s rational expectation assumption.

4 Estimation Results

In the following section, we present the estimated parameters and discuss the predicted first and second moments of endogenous variables. Additionally, we compare the historical components of the ten-year yield predicted by our model with estimates from the literature.

4.1 Parameter Estimates

As discussed in section 3.1, unlike standard perturbations (e.g. Andreasen et al., 2017), our solution method maintains linearity in states and shocks which allows us to use standard Bayesian techniques to estimate the model. In particular, we estimate the posterior mode of the distribution and employ a random walk Metropolis-Hasting algorithm to simulate the posterior distribution of the parameters and to quantify the uncertainty of our estimates

of the same. In particular, we run two chains, each with 100,000 parameter vector draws where the fist 50% have been discarded. Table3provides posterior statistics of the estimated parameters, e.g., the posterior mode, posterior mean and the 90% posterior credible set.¹¹ The results indicate that the posterior distributions of all structural parameters are well approximated and differ from the initial prior distribution. In the following, we discuss some key parameters in greater detail.

Name Symbol Mode Mean 5% 95%

Relative risk aversion RRA 89.860 91.427 75.581 108.489

Calvo parameter γp 0.853 0.855 0.843 0.866

Investment adjustment ν 1.417 1.440 1.204 1.667

Habit formation b 0.685 0.679 0.614 0.741

Intertemporal elas. substitution IES 0.089 0.089 0.077 0.101 Steady state inflation 100 (¯π−1) 1.038 1.034 0.981 1.091 Interest rate AR coefficient ρR 0.754 0.752 0.718 0.786 Interest rate inflation coefficient ηπ 3.124 3.164 2.839 3.491 Interest rate output coefficient ηy 0.156 0.159 0.114 0.204 Inflation target coefficient 100ζπ 0.210 0.242 0.109 0.366

AR coefficient technology ρ_a 0.366 0.356 0.304 0.412

AR coefficient preference ρ_b 0.820 0.817 0.793 0.843

AR coefficient investment ρ_i 0.956 0.955 0.949 0.961

AR coefficient gov. spending ρ_g 0.910 0.909 0.880 0.937 AR coefficient inflation target ρπ 0.934 0.925 0.901 0.950 AR coefficient long-run growth ρz 0.630 0.611 0.500 0.729

AR coefficient fixed cost ρΩ 0.928 0.928 0.922 0.933

S.d. technology 100σa 2.333 2.460 1.929 2.985

S.d. preference 100σb 4.878 4.880 4.180 5.570

S.d. investment 100σi 2.516 2.523 2.337 2.689

S.d. monetary policy shock 100σm 0.561 0.572 0.494 0.653 S.d. government spending 100σ_g 2.010 2.018 1.825 2.220

S.d. inflation target 100σ_π 0.167 0.180 0.130 0.226

S.d. long-run growth 100σ_z 0.345 0.353 0.253 0.446

S.d. fixed cost 100σ_Ω 9.766 9.705 9.022 10.372

ME 1-year T-Bill 400R^$_4,t 0.185 0.188 0.161 0.214

ME 2-year T-Bill 400R^$_8,t 0.084 0.085 0.071 0.100

ME 3-year T-Bill 400R^$_12,t 0.078 0.081 0.067 0.095

ME 5-year T-Bill 400R^$_20,t 0.152 0.156 0.130 0.181

ME 10-year T-Bill 400R^$_40,t 0.287 0.297 0.251 0.346

ME 1Q-expected policy rate 400Et

h R^f_t,t+1i

0.456 0.464 0.408 0.522 ME 4Q-expected policy rate 400Et

h R^f_t,t+4i

0.738 0.750 0.660 0.842

Table 3: Posterior statistics. Posterior means and parameter distributions are based on a standard MCMC algorithm with two chains of 100,000 parameter vector draws each, 50% of the draws used for burn-in, and a draw acceptance rates about 1/3.

11Figures10and11in the appendix illustrate the posterior distribution of each parameter in comparison to its initial prior distribution.